D.1 Stellar Quantities

Objects in the Universe

• The solar system includes the Sun, eight planets, dwarf planets, moons, asteroids, and comets.

• The universe is vast; our solar system is a mere speck.

Types of Celestial Bodies

• Single star: Luminous plasma sphere held by gravity.

• Binary star: Two stars orbiting a common center.

• Black hole: Singularity in space-time.

• Cepheid variable: Star with varying luminosity, aiding distance measurement.

• Clusters of galaxies: Gravitationally affected groups of galaxies.

• Constellation: Pattern of unbound stars visible from Earth.

• Dark matter: Cold, non-radiating matter inferred from physics.

• Galaxies: Stars, gas, and dust bound by gravity.

• Main sequence star: A normal star undergoing hydrogen fusion in order to turn into helium.

• Neutron stars: Dense stars with uncharged neutrons.

• Nebula: Cloud of dust, gasses, helium, and hydrogen.

• Planets: Celestial bodies orbiting a star.

• Supernova: Highly energetic stellar explosions marking the end of a star's life cycle.

  • Type Ia Supernova: Results from the explosion of a white dwarf in a binary star system.

  • Type Ib/c Supernova: Associated with the collapse of massive, hydrogen-poor stars.

  • Type II Supernova: Arises from the collapse of massive stars with a significant hydrogen envelope.

• White dwarfs: The remnants of low to medium-mass stars after they have exhausted their nuclear fuel.

The Nature of Stars

Stability and Equilibrium

• Star stability depends on the equilibrium between gravity and radiation pressure.

• Nuclear fusion maintains equilibrium, preventing collapse.

Units in Astrophysics

• Lightyear (ly): Distance light travels in one year in the vacuum of space. Approximately 9.461 × 10¹² kilometers.

• Parsec (pc): Parallax arcsecond, a unit based on stellar parallax. Approximately 3.09 × 10¹³ kilometers.

• Astronomical Unit (AU): Average distance from Earth to the Sun. Approximately 1.496 × 10⁸ kilometers.

• Megaparsec (Mpc): One million parsecs, often used in cosmological distance measurements. Approximately 3.09 × 10¹⁹ kilometers.

• Solar Radius (R☉): The radius of the Sun, used to express the size of stars. Approximately 6.96 × 10⁵ kilometers.

• Solar Mass (M☉): The mass of the Sun, frequently used for stellar mass comparisons. Approximately 1.989 × 10³⁰ kilograms.

• Light-Minute (lmin): Distance light travels in one minute. Approximately 1.8 × 10¹⁰ kilometers.

Astronomical Distances

• The universe is mostly empty; a light year measures ultra-solar system distances.

• Example: Proxima Centauri - 4.31 light years or 1.3 parsecs away.

• Average distance between stars in a galaxy: 1 pc (3.26 light-years).

• Average distance between galaxies in a cluster: 100 kpc to several Mpc.

Stellar Parallax and Limitations

• Stellar parallax measures space distances using Earth's orbit.

• The parallax of one arcsecond equals one parsec (3.26 light-years).

• There is limited accuracy for distant stars due to small parallax.

Luminosity and Apparent Brightness

• Luminosity: Total power radiated by a star in all directions (measured in watts).

• Apparent brightness: Power received per unit area (measured in W/m²).

• Luminosity decreases with distance following the inverse square law.

  • Inverse square law: I = k / r². States that a physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity. 

    • I = the intensity or strength of a physical quantity,

    • k = a constant

    • r = the distance from the source of the physical quantity.

D.2 Stellar Characteristics and Stellar Evolution

Stellar Spectra

• The absorption spectra can identify elements in stars.

• There are seven spectral classes (O, B, A, F, G, K, M) based on temperature. 

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Hertzsprung–Russell (HR) Diagram

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• It is a graph relating absolute magnitude, luminosity, classification, and temperature.

• Main sequence stars burn hydrogen; used to estimate star distances.

Mass–Luminosity Relation for Main Sequence Stars

• Luminosity increases with mass for main sequence stars.

Cepheid Variables

• Stars with varying luminosity correlated to period.

• Used as “standard candles” for distance estimation.

Stellar Evolution on HR Diagrams

• Stars form from nebulae, and then undergo nucleosynthesis.

• Main-sequence lifetime: Hydrogen fusion into helium.

• Red giants, white dwarfs, neutron stars, and black holes follow fuel depletion.

Chandrasekhar and Oppenheimer–Volkoff Limits

• Chandrasekhar limit: Maximum mass for a white dwarf (about 1.4 solar masses).

  Oppenheimer–Volkoff limit: Maximum mass for a neutron star (2-3 solar masses).

Wien’s Displacement Law

• Describes the relationship between the temperature of a blackbody and the wavelength at which it emits the maximum intensity of radiation.

• Mathematically expressed as λ_max ⋅T = constant where λ_max is the peak wavelength, and T is the temperature in Kelvin.

• Implies that as the temperature of a blackbody increases, the peak emission shifts to shorter (cooler) or longer (hotter) wavelengths.

• Crucial in understanding the color of stars; hotter stars appear bluer, while cooler stars appear redder.

D.3 Cosmology

Big Bang Model

• It is the origin of space and time from singularity expansion.

• It redshifted galaxy observation and Cosmic Microwave Background radiation support.

Cosmic Microwave Background (CMB) Radiation

• It is thermal radiation from the early universe, supporting the Big Bang theory.

Hubble’s Law

• v = Hd describes velocity-distance relationship.

• It is used to estimate the age of the universe.

• The Hubble Constant: denoted as H₀; quantifies the present rate of expansion of the universe, approximately 70 km/s/Mpc.

Accelerating Universe and Redshift (z)

• Supernovae observations show universe expansion acceleration.

• Redshift (z) is determined by the ratio of the observed (λ_observed) to (λ_emitted) emitted wavelengths, expressed as 1+z = (λ_observed / λ_emitted), or in cosmological contexts, z = (Δλ / λ_emitted) = (c⋅Δt)/(λ_emitted), where c is the speed of light and t is time.

• Redshift factor (1+z) affects apparent brightness.

Cosmic Scale Factor (R)

• The cosmic scale factor (R) is a fundamental concept in cosmology, serving as a mathematical representation of the relative expansion or contraction of the universe as a function of cosmic time.

• R is a dynamic parameter that evolves over time, capturing the changing size of the universe. As the universe expands, R increases, reflecting the overall growth of cosmic structures.

• R(t) represents the relative expansion of the universe.

• Einstein’s Theory of General Relativity: Astrophysicists employ Einstein's theory of general relativity to understand the behavior of R in the context of gravitational interactions on cosmic scales.

• Connection to Redshift: The concept of R is intimately connected to the observed redshift (z) in astrophysics. The relationship is expressed by 1+z= 1/R, offering a crucial link between observational data, such as the redshift of distant galaxies, and the underlying dynamics of the expanding universe.